Klein-Gordon Research Articles

Dynamical analysis of soliton structures for the nonlinear third-order Klein–Fock–Gordon equation under explicit approach

Dynamical analysis of soliton structures for the nonlinear third-order Klein–Fock–Gordon equation under explicit approach

Bilocal Field Theory for Composite Scalar Bosons.

We give a bilocal field theory description of a composite scalar with an extended binding potential that reduces to the Nambu-Jona-Lasinio (NJL) model in the pointlike limit. This provides a description of the internal dynamics of the bound state and features a static internal wave function, ϕ(r→), in the center-of-mass frame that satisfies a Schrödinger-Klein-Gordon equation with eigenvalues m2. We analyze the "coloron" model (single perturbative massive gluon exchange) which yields a UV completion of the NJL model. This has a BCS-like enhancement of its interaction, ∝Nc the number of colors, and is classically critical with gcritical remarkably close to the NJL quantum critical coupling. Negative eigenvalues for m2 lead to spontaneous symmetry breaking, and the Yukawa coupling of the bound state to constituent fermions is emergent.

Numerical solution of nonlinear Hunter-Saxton equation, Benjamin-Bona Mahony equation, and Klein-Gordon equation using Hosoya polynomial method

Models of nonlinear partial differential equations are essential in the theoretical sciences, especially Physics and Mathematics. It is exceedingly challenging to arrive at an analytical solution. For this reason, scientists are constantly looking for new computational techniques. For specific nonlinear mathematical models, such as the extensively researched Hunter-Saxton equation (HSE), Benjamin-Bona-Mahony equation (BBME), and Klein-Gordon equation (KGE), we are putting forth a novel and effective graph theoretic polynomial method named the Hosoya polynomial collocation method (HPCM). The orthonormalized Hosoya polynomials of the path graph and their operational matrices serve as the functional foundation for the new HPCM. The considered nonlinear models were transformed into a system of nonlinear algebraic equations using the operational matrix approximation and an appropriate collocation approach to reach the approximate solution. Six numerical examples verified the HPCM's productivity. Compared to the current numerical approaches in the literature, the results are significantly more efficient and almost match the analytical solution.

Global well-posedness of solutions for 2-D Klein–Gordon equations with exponential nonlinearity

This paper considers the global well-posedness of two-dimensional Klein–Gordon equations with exponential nonlinearity. By employing the potential well method, we conduct a comprehensive study on the global existence and finite time blowup of solutions by the requirement of the initial energy at three different initial energy levels.

Stabilization of the statistical solutions for large times for a harmonic lattice coupled to a Klein-Gordon field

Рассматривается задача Коши для гамильтоновой системы, состоящей из поля Клейна-Гордона и бесконечного гармонического кристалла. Динамика системы трансляционно-инвариантна относительно дискретной подгруппы $\mathbb{Z}^d$ группы $\mathbb{R}^d$. Начальные данные задаются как случайная функция, которая близка к двум пространственно-однородным (относительно подгруппы $\mathbb{Z}^d$) процессам при $\pm x_1>a$ с некоторым $a>0$. Изучается распределение $\mu_t$ решения в момент времени $t\in\mathbb{R}$ и доказывается слабая сходимость $\mu_t$ к гауссовой мере $\mu_\infty$ при $t\to\infty$. Кроме того, доказано существование предела корреляционных функций и выведены явные формулы для ковариации предельной меры $\mu_\infty$. Дается приложение к случаю гиббсовских мер.

Relativistic quantum Otto engine: instant work extraction from a quantum field

In this study, we carry out a non-perturbative approach to a quantum Otto engine, employing an Unruh-DeWitt particle detector to extract work from a quantum Klein-Gordon field in an arbitrary globally hyperbolic curved spacetime. We broaden the scope by considering the field in any quasi-free state, which includes vacuum, thermal, and squeezed states. A key aspect of our method is the instantaneous interaction between the detector and the field, which enables a thorough non-perturbative analysis. We demonstrate that the detector can successfully extract positive work from the quantum Otto cycle, even when two isochoric processes occur instantaneously, provided the detector in the second isochoric process receives a signal from the first interaction. This signaling allows the detector to release heat into the field, thereby the thermodynamic cycle is completed. As a demonstration, we consider a detector at rest in flat spacetime and compute the work extracted from the Minkowski vacuum state.

Symmetry Breaking and Dynamic Transition in the Negative Mass Term Klein–Gordon Equations

Through the discussion of three physical processes, we show that the Klein–Gordon equations with a negative mass term describe special dynamics. In the case of two classical disciplines—mechanics and thermodynamics—the Lagrangian-based mathematical description is the same, even though the nature of the investigated processes seems completely different. The unique feature of this type of equation is that it contains wave propagation and dissipative behavior in one framework. The dissipative behavior appears through a repulsive potential. The transition between the two types of dynamics can be specified precisely, and its physical meaning is clear. The success of the two descriptions inspires extension to the case of electrodynamics. We reverse the suggestion here. We create a Klein–Gordon equation with a negative mass term, but first, we modify Maxwell’s equations. The repulsive interaction that appears here results in a charge spike. However, the Coulomb interaction limits this. The charge separation is also associated with the high-speed movement of the charged particle localized in a small space domain. As a result, we arrive at a picture of a fast vibrating phenomenon with an electromagnetism-related Klein–Gordon equation with a negative mass term. The calculated maximal frequency value ω=1.74×1021 1/s.

On Klein–Gordon Gürses oscillators and pseudo-Gürses oscillators: Vorticity–energy correlations and spacetime-associated degeneracies

In this paper, we discuss Klein–Gordon (KG) oscillators in the [Formula: see text]-dimensional Gürses spacetime and under position-dependent mass (PDM) settings. We observe that the KG Gürses oscillators are introduced as a byproduct of the very nature of the Gürses spacetime structure. We report that the energy levels of such KG Gürses oscillators admit vorticity–energy correlations as well as spacetime-associated degeneracies (STADs). Moreover, we introduce a new set of KG pseudo-Gürses oscillators that admits isospectrality and invariance with the KG Gürses oscillators and inherits the same vorticity–energy correlations as well as STADs.

Exact massless scalar quasibound states of the Ernst black hole

In this work, we present a detailed derivation of novel exact massless scalar quasibound state a static magnetized Ernst black hole background. We successfully solve the governing covariant Klein–Gordon equation and discover the exact radial solutions in terms of the Confluent Heun functions. With the exact radial wave solution in hand, applying its polynomial condition leads to the discovery of the quantized energy levels expression that depends on the black hole’s mass M, magnetic field strength B0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_0$$\\end{document} and also the magnetic and main quantum number (mℓ,n).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m_\\ell , n).$$\\end{document} A massive scalar field around the black hole has complex valued energy levels while massless particle has purely imaginary energy levels. Further investigation in small black hole limit and zero magnetic field, the massless scalar’s purely imaginary energy expression is recovered. We also discover the equivalence between massless scalar field around an Ernst black hole with massive scalar field around a Schwarzschild black hole. In the last section, the Hawking radiation in investigated and applying the Damour–Ruffini method, the Hawking temperature is obtained out of the radiation distribution function.

Exact scalar quasibound states solutions of f(R) theory's static spherically symmetric black hole

Following the recent developments in black holes' quasibound states exact solutions by Senjaya (2023b,c), in this letter, we present a novel exact solution of quasibound states of a scalar around a static spherically symmetric black hole solution of f(R) theory of gravity. We start with the relativistic scalar field wave equation, i.e. the Klein-Gordon equation, isolating the radial equation and finally successfully obtain the exact radial wave solutions in terms of General Heun functions. Having the exact solutions in hand, the energy levels expression is obtained from the radial wave polynomial conditions. In the last part of this letter, the Hawking radiation is investigated via the Damour-Ruffini method and the Hawking temperature is obtained from radiation distribution function.

A possibility of Klein paradox in quaternionic (3 + 1) frame

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle’s kinetic energy is less than [Formula: see text]. Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero.

A conservative fully discrete finite element scheme for the nonlinear Klein–Gordon equation

This article proposes a family of the Petrov–Galerkin–FEM methods that can be used to solve the nonlinear Klein–Gordon equation. The discrete schemes were formulated based on the solution of the problem and its time derivative. They ensure that the total energy is conserved at a discrete level. The simplest two-layer scheme was studied numerically. Based on the solution of the test problems with smooth solutions, it was shown that the scheme can determine the solution of the problem, as well as its time derivative with an error of the order of O(h2 + τ 2) in the continuous L2 norm, where τ and h characterize the grid steps in time and space, respectively.

Comment on “Effects of (1+2)-dimensional circularly symmetric and static traversable wormhole with cosmic string on oscillator field”

We analyze the mathematical results and physical conclusions put forward in a recent paper about “the relativistic quantum dynamics of an oscillator field described by the Klein–Gordon equation in a circularly symmetric and static (1+2)-dimensional traversable wormhole spacetime with cosmic strings”. We show that the existence of allowed oscillator frequencies conjectured by the author is a mere artifact of the truncation of a power series with the purpose of obtaining bound-state solutions in terms of polynomials. Our theoretical analysis suggest that the eigenvalues are continuous, increasing functions of the oscillator frequency, conclusion that is verified by a straightforward numerical calculation.

Investigation of hidden symmetry in AdS5 black hole with Heun equation

Investigating the existence of algebra and finding hidden symmetries in physical systems is one of the most important aspects for understanding their behavior and predicting their future. Expanding this unique method of study to cosmic structures and combining past knowledge with new data can be very interesting and lead to discovering new ways to analyze these systems. However, studying black hole symmetries always presents many complications and sometimes requires computational approximations. For example, checking the existence of Killing vectors and then calculating them is not always an easy task. It becomes much more difficult as the structure and geometry of the system become more complex. In this work, we will show that if the wave equations with a black hole background can be converted in the form of general Heun equation, based on its structure and coefficients, the algebra of the system can be easily studied, and computational and geometrical complications can be omitted. For this purpose, we selected two [Formula: see text] black holes: Reissner–Nordstrm (RN) and Kerr, and analyzed the Klein–Gordon equation with the background of these black holes. Based on this concept, we observed that the radial part of the RN black hole and both the radial and angular parts of the Kerr black hole could be transformed into the general form of the Heun equation. As a result, according to the algebraic structure that governs the Heun equation and its coefficients, one can easily achieve generalized sl(2) algebra.

Behavior of spin-0 scalar particles in Eddington-inspired Born–Infeld gravity global monopole with harmonic oscillator potential

In this paper, we investigate the behavior of a scalar bosonic field within the backdrop of Eddington-inspired Born–Infeld gravity with a global monopole configuration. Additionally, we consider the presence of both scalar and vector potentials. To introduce the scalar potential, we perform a transformation on the mass term, replacing [Formula: see text] with [Formula: see text]. For the vector potential, we apply a minimal coupling scheme, transforming [Formula: see text] within the relativistic Klein–Gordon equation. We opt for a harmonic oscillator scenario where the scalar and vector potentials are equal, i.e. [Formula: see text]. Through this setup, we analytically solve the wave equation using the confluent Heun equation form. Subsequently, we delve into the quantum system in the absence of any potential, determining the permissible values for bound-state relativistic energy levels and the scalar field’s wave function. Our analysis reveals that the energy levels and wave function are influenced by various parameters present in the eigenvalue expression.

Existence of solutions for critical Klein–Gordon equations coupled with Born–Infeld theory in higher dimensions

In this paper, we investigate the existence of nontrivial solutions for nonlinear Klein–Gordon equations coupled with Born–Infeld theory with critical Sobolev exponents by variational methods.

De Broglie, General Covariance and a Geometric Background to Quantum Mechanics

What is striking about de Broglie’s foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ2Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow.

Modulation Instability, Analytical, and Numerical Studies for Integrable Time Fractional Nonlinear Model through Two Explicit Methods

The present work deals with the investigation of the time‐fractional Klein–Gordon (K‐G) model, which has great importance in theoretical physics with applications in various fields, including quantum mechanics and field theory. The main motivation of this work is to analyze modulation instability and soliton solution of the time‐fractional K‐G model. Comparative studies are investigated by β‐fraction derivative and M‐fractional derivative. For this purpose, we used unified and advanced exp(−ϕ(ξ))‐expansion approaches that are highly important tools to solve the fractional model and are used to create nonlinear wave pattern (both solitary and periodic wave) solutions for the time‐fractional K‐G model. For the special values of constraints, the periodic waves, lumps with cross‐periodic waves, periodic rogue waves, singular soliton, bright bell shape, dark bell shape, kink and antikink shape, and periodic wave behaviors are some of the outcomes attained from the obtained analytic solutions. The acquired results will be useful in comprehending the time‐fractional K‐G model’s dynamical framework concerning associated physical events. By giving specific values to the fractional parameters, graphs are created to compare the fractional effects for the β‐fraction derivative and M‐fractional derivative. Additionally, the modulation instability spectrum is expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third‐order dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to theoretical physics. This work provided the effect of the recently updated two fraction forms, and in the future, we will integrate the space–time M‐fractional form of the governing model by using the extended form of the Kudryashov method. Maple 18 is utilized as the simulation tool.

IMPLEMENTATION AND ASSESSMENT OF THE SIMPLE EQUATION TECHNIQUE FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

In this paper, the simple equation method is especially used to solve two Nonlinear Partial Differential Partial Equations NLPDEs, the Kodomstev-Petviashvili (KP) equation and the (2+1)-dimensional breaking soliton equation. The modified Benjamin-Bona-Mahony equation and the Klein-Gordon equation in (1+2) dimensions are two illustrations of second order nonlinear equations that can benefit from using this approach. The Bernoulli equation acts as the trial condition and aids in the mathematical a description of the nonlinear wave equation in the simple equation.

Rigorous a Posteriori Error Bounds for PDE-Defined PINNs.

Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks (NNs), for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of NNs, we present a rigorous upper bound on the prediction error of physics-informed NNs (PINNs). This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation (PDE). We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation (NSE), and the Klein-Gordon equation.